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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > ltrniotaidvalN | Structured version Visualization version GIF version |
Description: Value of the unique translation specified by identity value. (Contributed by NM, 25-Aug-2014.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ltrniotaidval.b | ⊢ 𝐵 = (Base‘𝐾) |
ltrniotaidval.l | ⊢ ≤ = (le‘𝐾) |
ltrniotaidval.a | ⊢ 𝐴 = (Atoms‘𝐾) |
ltrniotaidval.h | ⊢ 𝐻 = (LHyp‘𝐾) |
ltrniotaidval.t | ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) |
ltrniotaidval.f | ⊢ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) |
Ref | Expression |
---|---|
ltrniotaidvalN | ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐹 = ( I ↾ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltrniotaidval.l | . . . 4 ⊢ ≤ = (le‘𝐾) | |
2 | ltrniotaidval.a | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | ltrniotaidval.h | . . . 4 ⊢ 𝐻 = (LHyp‘𝐾) | |
4 | ltrniotaidval.t | . . . 4 ⊢ 𝑇 = ((LTrn‘𝐾)‘𝑊) | |
5 | ltrniotaidval.f | . . . 4 ⊢ 𝐹 = (℩𝑓 ∈ 𝑇 (𝑓‘𝑃) = 𝑃) | |
6 | 1, 2, 3, 4, 5 | ltrniotaval 39035 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐹‘𝑃) = 𝑃) |
7 | 6 | 3anidm23 1421 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐹‘𝑃) = 𝑃) |
8 | simpl 483 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻)) | |
9 | 1, 2, 3, 4, 5 | ltrniotacl 39033 | . . . 4 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐹 ∈ 𝑇) |
10 | 9 | 3anidm23 1421 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐹 ∈ 𝑇) |
11 | simpr 485 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) | |
12 | ltrniotaidval.b | . . . 4 ⊢ 𝐵 = (Base‘𝐾) | |
13 | 12, 1, 2, 3, 4 | ltrnideq 38629 | . . 3 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ 𝐹 ∈ 𝑇 ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐹 = ( I ↾ 𝐵) ↔ (𝐹‘𝑃) = 𝑃)) |
14 | 8, 10, 11, 13 | syl3anc 1371 | . 2 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → (𝐹 = ( I ↾ 𝐵) ↔ (𝐹‘𝑃) = 𝑃)) |
15 | 7, 14 | mpbird 256 | 1 ⊢ (((𝐾 ∈ HL ∧ 𝑊 ∈ 𝐻) ∧ (𝑃 ∈ 𝐴 ∧ ¬ 𝑃 ≤ 𝑊)) → 𝐹 = ( I ↾ 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 class class class wbr 5105 I cid 5530 ↾ cres 5635 ‘cfv 6496 ℩crio 7311 Basecbs 17082 lecple 17139 Atomscatm 37716 HLchlt 37803 LHypclh 38438 LTrncltrn 38555 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7671 ax-riotaBAD 37406 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rmo 3353 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-iin 4957 df-br 5106 df-opab 5168 df-mpt 5189 df-id 5531 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7312 df-ov 7359 df-oprab 7360 df-mpo 7361 df-1st 7920 df-2nd 7921 df-undef 8203 df-map 8766 df-proset 18183 df-poset 18201 df-plt 18218 df-lub 18234 df-glb 18235 df-join 18236 df-meet 18237 df-p0 18313 df-p1 18314 df-lat 18320 df-clat 18387 df-oposet 37629 df-ol 37631 df-oml 37632 df-covers 37719 df-ats 37720 df-atl 37751 df-cvlat 37775 df-hlat 37804 df-llines 37952 df-lplanes 37953 df-lvols 37954 df-lines 37955 df-psubsp 37957 df-pmap 37958 df-padd 38250 df-lhyp 38442 df-laut 38443 df-ldil 38558 df-ltrn 38559 df-trl 38613 |
This theorem is referenced by: dihpN 39790 |
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